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Vol. 5
COMPETITION QUOTIENT IN YOUNG PINUS RADIATA
R. B. T E N N E N T
Forest Research Institute, New Zealand Forest Service, Rotorua
(Received for publication 1 August 1974)
For the construction of yield models on an individual tree basis a measure of the
competitive stress experienced by an individual tree is required. Many such measures
have been suggested. Brown's (1965) Area Potentially Available (APA) is an example.
It is in effect the inverse of the stocking at the point occupied by the subject tree. The
method is objective, as no decision needs to be made on the competitive effect of
neighbours. But the APA does not account for differences in tree size, and so fails
to account for difference in competitive ability of neighbours.
A different family of competition indices has been developed since the advent
of electronic data processing, based on Staebler's (1951) premise. Staebler's index was
based on the assumption that the growing space occupied by an individual tree is a
circular area whose radius is related to the tree's diameter at breast height (d.b.h.^) by
a (usually) linear function. Staebler hypothesised that the area of overlap of the
competition circles so constructed for adjacent trees would supply a direct measure
of competition (see Fig. 1). As the mathematical formulae which measure the overlap
FIG. 1—The overlapping of competition circles of adjacent trees.
N.Z. J. For. Sci. 5 (2): 230-4 (1975).
No. 2
NOTE
231
area are relatively complex for manual calculation Staebler adopted the linear overlap
(i.e. the radial width of the overlap region) as his index.
Intrinsic to all such indices is the function used to define the competition circles.
Various attempts to model physical conditions have been made. The competition circle
radius has been related to rooting area radius, and, most commonly, the radius of opengrown crowns. Bella (1969) and others have used functions which relate the crown
radius of open-grown trees to d.b.h., e.g.
CR = 1.815 + .805 D
where CR = crown radius (ft)
D = d.b.h. (in.)
The competition circle is then defined by the open-grown crown radius the subject
tree would have had, had it been open-grown. However, such methods overlook the
fact that a stand-grown tree with a similar d.b.h. to an open-grown tree may have
already had its d.b.h. growth restricted through competition, and as such its hypothetical
"open-grown crown radius" bears little relationship to its competition zone.
Gerrard (1969) developed a similar index which is related to the subject tree's
size. His revision of Staebler's premise was:
"The competitive stress sustained by a tree is directly proportional to the area
of overlap of its competition circle with those of its neighbours and inversely
proportional to the area of its own competition circle."
This in effect assumes that the larger the tree the more intense the competition k
should be able to endure. His index, termed competition quotient, is defined by:
1 n
- 2 ak
A k=1
where n = number of competitors
ak = area of overlap of the kth competitor
A = competition circle area of the subject tree
Gerrard's index differs little from the general family of indices, except for the
use of the actual overlap area.* However, his study was of interest because of the
manner in which he defined his competition circle. He reasoned that the constant
(termed competition radius factor) which when multiplied by a tree's d.b.h. gave the
most significant measure of competition should be used for any future analysis. He
examined a range of competition radius factors to discover the optimum competition
radius factor for a variety of species in a mixed-age, mixed species stand on a varying
site. He concluded: ". . . further investigations . . . should be directed at pure stands,
preferably plantations, wherein the factors controlling the performance of individual
trees are far less numerous and more susceptible to measurement".
Competition Quotient =
Examination of the effectiveness of competition quotient was conducted using data
collected from a Nelder design spacing trial, analysed elsewhere (Tennent, 1973). The
trial was used as it was considered to suit the recommendations made by Gerrard. It
is a pure Pinus radiata planted trial, occupying 1 ha on a uniform site. The planting
* J. E. Opie (1968) has independently developed a similar index, differing mainly in
definition.
232
New Zealand Journal of Forestry Science
Vol. 5
spacings vary uniformly from l X l m t o 4 x 4 m . D.b.h. vary from 3 to 18 cm, and
data were available for two increment periods, age 4 to 5 years, and age 5 to 7. The
top height at age 7 was 10 m.
The investigation followed Gerrard's closely. A second order model was constructed
and the analysis run at different competition radius factors. The model used is given
below:
DI = b 0 + biD + b 2 D 2 + b 3 C + b 4 C 2 + b 5 C X D
(1)
where DI = mean annual d.b.h. increment (cm)
D = d.b.h. at start of growth period (cm)
C = competition quotient
bi = regression coefficient.
The model is not claimed to be anything other than a first approximation of the
relationship between diameter increment, diameter and competition quotient. All terms
were included regardless of their significance. The analysis consisted of varying the
competition radius factor in the assessment of competition quotient over a range of
.15 to .31 for the age 4 to 5 increment period, and from .1 to .35 for the age 5 to 7
increment period. The analysis was also conducted on an abbreviated model,
DI = b 0 + biD + b 2 D 2
to examine the effectiveness of competition quotient.
(2)
Figs. 2 and 3 show the effect of varying the competition radius factor on the
coefficient of multiple determination, R 2 , of the full model for the growth periods
age 4 to 5 and age 5 to 7 respectively.
0-7 r-
LU
-J
DL
• CN
O H
H 2 0-6 h
UJ
o
er
LU
hLU
LU Q
o
o
0-5
0-15
0-20
0-25
0-30
COMPETITION RADIUS FACTOR
FIG. 2—The effect of varying competition radius factors on the significance of competition
quotient for the growth period age 4 to 5 years.
233
Note
No. 2
CS
or
0-8
<
er
LU
LU
Q 0-7
LU
O 0-6
LU
o
LU
o
o 0-5
JL
0-25
-L
0-20
0-15
_L
0-30
0-35
COMPETITION RADIUS FACTOR
FIG. 3—The effect of varying competition radius factors on the significance of competition
quotient for the growth period age 5 to 7 years.
Table 1 shows the standard error and R2 values for equation 2, the abbreviated
model, and equation 1, the full model, with a competition radius factor of .21 for
both increment periods.
The asymptotic maximum reached in both increment periods can be explained as
follows. Intuitively the most suitable competition radius factor would be that which
results in no overlapping of competition circles for trees which are not under competiTABLE 1—The improvement in regression significance from the addition of competition
quotient
Equation 2
Equation 1
Age at
start of
growth
period
Standard
error
R2
Standard
error
R2
4
1.010
0.055
0.671
0.584
1102
5
0.958
0.275
0.497
0.805
1096
N
234
New Zealand Journal of Forestry Science
Vol. 5
tive stress. However, the effect of increasing the competition radius factor over this
level is to increase the magnitude of the competition quotient in all cases. As the
regression analysis relies on the relative magnitude of the competition quotient term,
and not the absolute magnitude, the effect on the R 2 value becomes minimal. The
process can be likened to the scaling of a variable in an analysis, which changes the
value of the regression coefficients, but not the significance of the actual regression.
This has an important consequence for future work. As the analysis at this stage
involves only two increment periods in young stands the findings are tentative, as a
time-dependent series of optimal competition radius factors is likely. However, the
apparent asymptotic nature of the R 2 analysis indicates that such a series would not
present a problem. The question is only to define the largest competition radius factorrequired over the life of the tree species. This competition radius factor could then be
used for all ages of the species, as the asymptotic nature of the analysis ensures that
a significant competition relationship can be found.
In both increment periods studied the asymptotic maximum is reached at a competition radius factor slightly above .2. Indications are that a competition radius factor
of .21 would be suitable for any further analysis. This would ensure that competition
is accounted for, and also prevent the boundary of the analysis of the competition, of
any individual tree becoming too wide.
The effectiveness of competition quotient in accounting for differences in diameter
increment can be judged from the dramatic increase in the value of R 2 and drop in
the standard error of the model (shown in Table 1) with the addition of the last three
terms in Equation 1. For the second growth period 8 0 % of the variation in diameter
increment can be explained by initial d.b.h. and competition quotient. A more vigorous
model could no doubt account for more.
CONCLUSION
Examination has shown that competition quotient is a useful concept for analysis
of the competition experienced by individual trees. The effectiveness of the index is
dependent on the competition radius factor used to define the competition circles of
each tree. At present no firm statement of a suitable competition radius factor for
Pinus radiata can be made, but indications are that a competition radius factor of .21
may be acceptable for stands which have experienced the onset of competition.
Repetition of the analysis at a later stage will provide additional information.
REFERENCES
BELLA, I. E. 1989: Competitive influence—Zone overlap: a competition model for individual
trees. Dept. Fisheries and Forestry, Canada. Bi-m. Res. Notes 25 (3): 24-5.
BROWN, G. S. 1965: Point density in stems per acre. N.Z. For. Serv. Tech. Pap. No. 38.
11 PP.
GERRARD, D. J. 1969: Competition quotient: A new measure of the competition affecting
individual forest trees. Michigan State Univ. Agric. Exp. Sta. Res. Bull. No. 20.
30 pp.
OPIE, J. E. 1968: Predictability of individual tree growth using various definitions of
competing basal area. For. Sci., 14 (3): 314-23.
STAEBLER, G. R. 1951: Growth and spacing in an even-aged stand of Douglas fir. Michigan
Univ. unpubl. M.F. thesis.
TENNENT, R. B. 1973: Preliminary analysis of Nelder spacing trials at Kaingaroa Forest.
N.Z. For. Ser., For. Res. Inst., Economics of Silviculture Rep. No. 70 (unpubl.).